47-49] CYCLIC REGIONS. 55 



circuits in one point only, and not meeting any of the n 2 

 remaining circuits. A barrier drawn in this manner does not 

 destroy the continuity of the region, for the interrupted circuit 

 remains as a path leading round from one side to the other. The 

 order of connection of the region is however diminished by unity ; 

 for every circuit drawn in the modified region must be reconcileable 

 with one or more of the n 2 circuits riot met by the barrier. 



A second barrier, drawn in the same manner, will reduce the 

 order of connection again by one, and so on ; so that by drawing 

 n I barriers we can reduce the region to a simply-connected one. 



A simply-connected region is divided by a barrier into two 

 separate parts ; for otherwise it would be possible to pass from a 

 point on one side the barrier to an adjacent point on the other side 

 by a path lying wholly within the region, which path would in the 

 original region form an irreducible circuit. 



Hence in an n-ply- connected region it is possible to draw u 1 

 barriers, and no more, without destroying the continuity of the 

 region. This property is sometimes adopted as the definition of 

 an n-ply-connected space. 



Irrotational Motion in Multiply-connected Spaces. 



49. The circulation is the same in any two reconcileable 

 circuits AEG A, A B C A drawn in a region occupied by fluid 

 moving irrotationally. For the two circuits may be connected by 

 a continuous surface lying wholly within the region ; and if we 

 apply the theorem of Art. 33 to this surface, we have, remembering 

 the rule as to the direction of integration round the boundary, 



/ (ABC A) + 1 (A C B A ) = 0, 

 or / (ABCA) = I (A B C A }. 



If a circuit ABC A be reconcileable with two or more circuits 

 A B C A , A&quot;B&quot;C&quot;A&quot;, &c., combined, we can connect all these 

 circuits by a continuous surface which lies wholly within the 

 region, and of which they form the complete boundary. Hence 



/ (ABCA) + 1 (A C B A ) + 1 (A&quot;C&quot;B&quot;A&quot;) + &c. = 0, 

 or / (ABCA) = I (A B C A ) + 1 (A&quot;B&quot;C&quot;A&quot;) + &c. ; 

 i.e. the circulation in any circuit is equal to the sum of the 



